PCTRRFS(l)

📦 version 1.5, in ubuntu jammy

💖 PCTRRFS(l) — the LAPACK girlies' guide to error bounds 💖

okay bestie, so this is a LAPACK routine, version 1.5, and it's giving math girlboss. it's literally called PCTRRFS and it's here to hand you error bounds and backward error estimates for a linear system whose coefficient matrix is triangular. no cap. ✨

🌸 NAME 🌸

PCTRRFS — provide error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. periodt. 💅

✨ SYNOPSIS ✨

SUBROUTINE PCTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, IA, JA, DESCA, B, IB,
                    JB, DESCB, X, IX, JX, DESCX, FERR, BERR, WORK,
                    LWORK, RWORK, LRWORK, INFO )

    CHARACTER       DIAG, TRANS, UPLO

    INTEGER         INFO, IA, IB, IX, JA, JB, JX, LRWORK, LWORK, N, NRHS

    INTEGER         DESCA( * ), DESCB( * ), DESCX( * )

    REAL            BERR( * ), FERR( * ), RWORK( * )

    COMPLEX         A( * ), B( * ), WORK( * ), X( * )

🎀 PURPOSE 🎀

PCTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix.

okay so here's the vibe: the solution matrix X must already be computed by PCTRTRS or by some other means before you walk in here. and lowkey important — PCTRRFS does not do iterative refinement, because doing so cannot improve the backward error. so don't ask her to, she won't. 🫶

🦋 Notes 🦋

Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

here's the descriptor lore, bestie, spelled out for you:

  • DTYPE_A(global) — stored in DESCA( DTYPE_ ) — The descriptor type. In this case, DTYPE_A = 1.
  • CTXT_A (global) — stored in DESCA( CTXT_ ) — The BLACS context handle, indicating the BLACS process grid A is distributed over. The context itself is global, but the handle (the integer value) may vary.
  • M_A (global) — stored in DESCA( M_ ) — The number of rows in the global array A.
  • N_A (global) — stored in DESCA( N_ ) — The number of columns in the global array A.
  • MB_A (global) — stored in DESCA( MB_ ) — The blocking factor used to distribute the rows of the array.
  • NB_A (global) — stored in DESCA( NB_ ) — The blocking factor used to distribute the columns of the array.
  • RSRC_A (global) — stored in DESCA( RSRC_ ) — The process row over which the first row of the array A is distributed.
  • CSRC_A (global) — stored in DESCA( CSRC_ ) — The process column over which the first column of the array A is distributed.
  • LLD_A (local) — stored in DESCA( LLD_ ) — The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q.

LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.

Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

An upper bound for these quantities may be computed by:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

In the following comments, sub( A ), sub( X ) and sub( B ) denote respectively A(IA:IA+N-1,JA:JA+N-1), X(IX:IX+N-1,JX:JX+NRHS-1) and B(IB:IB+N-1,JB:JB+NRHS-1).

👑 ARGUMENTS 👑

every single one, bestie, no skips:

  • UPLO — (global input) CHARACTER*1
    • = 'U': sub( A ) is upper triangular;
    • = 'L': sub( A ) is lower triangular.
  • TRANS — (global input) CHARACTER*1 — Specifies the form of the system of equations.
    • = 'N': sub( A ) * sub( X ) = sub( B ) (No transpose)
    • = 'T': sub( A )**T * sub( X ) = sub( B ) (Transpose)
    • = 'C': sub( A )**H * sub( X ) = sub( B ) (Conjugate transpose)
  • DIAG — (global input) CHARACTER*1
    • = 'N': sub( A ) is non-unit triangular;
    • = 'U': sub( A ) is unit triangular.
  • N — (global input) INTEGER — The order of the matrix sub( A ). N >= 0.
  • NRHS — (global input) INTEGER — The number of right hand sides, i.e., the number of columns of the matrices sub( B ) and sub( X ). NRHS >= 0.
  • A — (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1) ). This array contains the local pieces of the original triangular distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distributed matrix, and its strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1.
  • IA — (global input) INTEGER — The row index in the global array A indicating the first row of sub( A ).
  • JA — (global input) INTEGER — The column index in the global array A indicating the first column of sub( A ).
  • DESCA — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
  • B — (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_B, LOCc(JB+NRHS-1) ). On entry, this array contains the the local pieces of the right hand sides sub( B ).
  • IB — (global input) INTEGER — The row index in the global array B indicating the first row of sub( B ).
  • JB — (global input) INTEGER — The column index in the global array B indicating the first column of sub( B ).
  • DESCB — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix B.
  • X — (local input) COMPLEX pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1) ). On entry, this array contains the the local pieces of the solution vectors sub( X ).
  • IX — (global input) INTEGER — The row index in the global array X indicating the first row of sub( X ).
  • JX — (global input) INTEGER — The column index in the global array X indicating the first column of sub( X ).
  • DESCX — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X.
  • FERR — (local output) REAL array of local dimension LOCc(JB+NRHS-1). The estimated forward error bounds for each solution vector of sub( X ). If XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (sub( X ) - XTRUE) divided by the magnitude of the largest entry in sub( X ). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. This array is tied to the distributed matrix X.
  • BERR — (local output) REAL array of local dimension LOCc(JB+NRHS-1). The componentwise relative backward error of each solution vector (i.e., the smallest relative change in any entry of sub( A ) or sub( B ) that makes sub( X ) an exact solution). This array is tied to the distributed matrix X.
  • WORK — (local workspace/local output) COMPLEX array, dimension (LWORK). On exit, WORK(1) returns the minimal and optimal LWORK.
  • LWORK — (local or global input) INTEGER — The dimension of the array WORK. LWORK is local input and must be at least LWORK >= 2*LOCr( N + MOD( IA-1, MB_A ) ).

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

  • RWORK — (local workspace/local output) REAL array, dimension (LRWORK). On exit, RWORK(1) returns the minimal and optimal LRWORK.
  • LRWORK — (local or global input) INTEGER — The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= LOCr( N + MOD( IB-1, MB_B ) ).

If LRWORK = -1, then LRWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

  • INFO — (global output) INTEGER
    • = 0: successful exit
    • < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i.

💀 Notes 💀

This routine temporarily returns when N <= 1. yes really, she just dips. 🙈

The distributed submatrices sub( X ) and sub( B ) should be distributed the same way on the same processes. These conditions ensure that sub( X ) and sub( B ) are "perfectly" aligned.

Moreover, this routine requires the distributed submatrices sub( A ), sub( X ), and sub( B ) to be aligned on a block boundary, i.e., if f(x,y) = MOD( x-1, y ): f( IA, DESCA( MB_ ) ) = f( JA, DESCA( NB_ ) ) = 0, f( IB, DESCB( MB_ ) ) = f( JB, DESCB( NB_ ) ) = 0, and f( IX, DESCX( MB_ ) ) = f( JX, DESCX( NB_ ) ) = 0.

fr fr bestie, get that alignment right or she will not be happy with you. 💅

okay that's the whole page, you ate that. this was LAPACK version 1.5, dated 12 May 1997 — yes she's an elder icon and she still slays. bye bestie, go get those error bounds! 🦋✨💖

📦 where this page comes from